Subgroups — C₄×C₇×D₄ #6521

Order 224.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₇	C₂	1	2	yes	maximal, complemented
3	C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₇	C₂²	1	2	yes	—
7	C₇×D₄	C₄	1	4	yes	direct product factor
8	C₂×C₄×C₇	—	2	4	no	—
9	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇	C₂²	1	2	yes	—
11	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₄×D₄	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₇	—	2	2	no	—
15	C₂²×C₇	—	2	2	no	—
16	C₂²×C₇	C₂×C₄	1	4	yes	complemented
17	C₂²×C₇	C₂³	1	1	yes	characteristic
18	C₄×C₇	—	2	2	no	—
19	C₄×C₇	—	2	1	no	—
20	C₄×C₇	C₂×C₄	1	2	yes	complemented
21	C₄×C₇	D₄	1	2	yes	direct product factor
22	C₂×D₄	C₂×C₇	1	1	yes	characteristic
23	C₂²×C₄	C₂×C₇	1	2	yes	complemented
24	C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
25	C₂²⋊C₄	C₂×C₇	1	2	yes	complemented
26	C₄²	C₂×C₇	1	1	yes	characteristic, complemented
27	C₂×C₇	—	2	4	no	—
28	C₂×C₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₇	#41	1	1	yes	characteristic
31	C₂³	C₂²×C₇	1	2	yes	—
32	D₄	C₄×C₇	1	4	yes	direct product factor
33	C₂×C₄	—	2	4	no	—
34	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
35	C₂×C₄	C₂²×C₇	1	2	yes	—
36	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
37	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
38	C₇	C₄×D₄	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²	—	2	2	no	—
40	C₂²	—	2	2	no	—
41	C₂²	C₂×C₄×C₇	1	4	yes	complemented
42	C₂²	C₂³×C₇	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	2	1	no	—
45	C₄	C₂×C₄×C₇	1	2	yes	complemented
46	C₄	C₇×D₄	1	2	yes	direct product factor
47	C₂	—	2	4	no	—
48	C₂	C₂²×C₄×C₇	1	1	yes	characteristic
49	C₂	C₂×C₇×D₄	1	1	yes	characteristic
50	C₂	C₇×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²
commutator	C₂	C₂²×C₄×C₇
Frattini, cosocle	C₂²	C₂³×C₇
socle	C₂²×C₇	C₂³
prime radical	C₂×C₇×D₄	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×C₄×C₇	C₂	semidirect
3. C₇×C₄⋊C₄	C₂	semidirect
4. C₇×C₂²⋊C₄	C₂	semidirect
5. C₄²×C₇	C₂	semidirect
7. C₇×D₄	C₄	direct (N × H)
13. C₄×D₄	C₇	direct (N × H)
16. C₂²×C₇	C₂×C₄	semidirect
20. C₄×C₇	C₂×C₄	semidirect
21. C₄×C₇	D₄	direct (N × H)
23. C₂²×C₄	C₂×C₇	semidirect
24. C₄⋊C₄	C₂×C₇	semidirect
25. C₂²⋊C₄	C₂×C₇	semidirect
26. C₄²	C₂×C₇	semidirect
32. D₄	C₄×C₇	direct (N × H)
38. C₇	C₄×D₄	direct (N × H)
41. C₂²	C₂×C₄×C₇	semidirect
45. C₄	C₂×C₄×C₇	semidirect
46. C₄	C₇×D₄	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×D₄	2
C₂²×C₄×C₇	2
C₇×C₄⋊C₄	2
C₇×C₂²⋊C₄	2
C₄²×C₇	2
C₄×D₄	7

Subgroups — C4×C7×D4 #6521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₇×D₄ #6521